An ideally efficient heat pump delivers 1000 J of heat to room air at 300 K. If it extracted heat from 260 K outdoor air, how much of that delivered heat was originally work consumed in the transfer?

Wnet, in, = 133.33J

Explanation:

Given that

Pump heat QH = 1000J

Warm temperature TH = 300K

Cold temperature TL = 260K

Since the heat pump is completely reversible, the combination of coefficient of performance expression is given as,

From first law of thermodynamics,

COP (HP, rev) = 1 / (1-TL/TH)

COP (HP, rev) = 1 / (1-260/300)

COP (HP, rev) = 1 / (1-0.867)

COP (HP, rev) = 1/0.133

COP (HP, rev) = 7.5

The power required to drive the the heat pump is given as

Wnet, in = QH/COP (HP, rev)

Wnet, in = 1000/7.5

Wnet, in = 133.333J. QED

So the 133.33J was the amount heat that was originally work consumed in the transfer.

Extra ...

According to the first law, the rate at which heat is removed from the low temperature reservoir is given as

QL=QH-Wnet, in

QL=1000-133.333

QL=866.67J